WASHINGTON — Republican senators on Tuesday challenged efforts to pass new U.S. gun laws, saying at a Judiciary Committee hearing that the government should work on enforcing existing regulations.

Freshman Sen. Ted Cruz of Texas was the first of several Republicans at the hearing to say Congress shouldn’t focus on advancing legislation that limits access to military-style assault weapons and the high-capacity magazines used in many recent mass shootings.

“Stripping the constitutional rights of law-abiding citizens does nothing to prevent criminals from carrying out a violent crime,” Cruz said at the panel’s second hearing on gun violence since the Dec. 14 shootings in Newtown, Conn., in which 20 children and six adults were killed. “Law enforcement has been failing. We should be working to fix that problem.”

As Congress debates new gun restrictions, Republicans put down markers before President Barack Obama delivered his State of the Union speech Tuesday night, during which he was expected to underscore his own proposals.

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“We’ve got our priorities wrong,” Sen. Lindsey Graham, R-S.C., said at the hearing. “We should take the current law and enforce it.”

Sen. John Cornyn of Texas said “we need to look at laws that are already on the books.”

In the face of such resistance, advocates for stricter gun laws are pressing for enough bipartisan support to advance new laws on background checks.

A number of Democrats at the hearing and U.S. Attorney Timothy Heaphy of Virginia, the main witness to testify, pushed for legislation that would expand background checks for gun buyers to keep weapons from criminals and the mentally ill.

House Democrats last week made recommendations that mirror Obama’s gun control proposals, including a ban on assault weapons and a 10-round limit on ammunition magazines. They also called for background checks for almost all gun buyers and for requiring states to include more mental health information in the national criminal background database.

The U.S. needs to “provide reasonable restrictions to keep dangerous weapons” out of the hands of criminals, Heaphy said. “Those acts from their past remove their Second Amendment right.”

At the same time, he said, “I can’t reiterate strongly enough” that the Second Amendment rights of law-abiding citizens are under no threat.

Democrats on the panel spoke in support of restrictions, especially tightening background checks for gun buyers.

“We need to improve and expand our background check system,” said Sen. Al Franken of Minnesota. “There’s a consensus being built around the idea that we should do a background check on all purchases of guns.”

Heaphy cited an exception that allows some purchasers at private gun shows to avoid background checks.

“That is a gaping hole in the system,” he said. “We have to do what we can on the front end” to prevent criminals from gaining access to guns.

Attending the hearing were the families of victims of gun violence, some of whom said they were disappointed with lawmakers’ response after the shootings at Newtown and elsewhere that gained national attention.

“They’re trying to make an issue that is so personal to us political,” said Tiffany Rice, the mother of a 14-year-old boy, Dajae Coleman, who was killed by gunfire walking home from a party in Evanston, Ill., in September. “I don’t understand why it’s so hard” to pass laws to limit criminal access to guns, she said.

Some House Republicans, many from urban areas and states such as Virginia that have been sites of mass shootings, are showing a willingness to thwart the National Rifle Association’s opposition to broadening background checks for gun purchases. A Quinnipiac University poll released on Feb. 7 found more than nine in 10 Americans support universal background checks.